If you file a comprehensive claim, it is more likely than not, that your rates will be higher than they would be if you do not file the claim. This may be done via tier placement or some other mechanism that is invisible to the consumer. When switching carriers, your CLUE report will show the claim, and it would be considered a Not at fault(NAF) claim by many carriers, so they would quote you a higher premium, especially if you have more than one NAF. Geico has one of the most sophisticated rating algorithms out there, and much of it is complicated in order to obscure what they are doing to prevent competitors from copying them, as well as to prevent insureds from understanding how their rates are impacted.
That said, themagicman is correct that in some states surcharging is not allowed on some or all comprehensive claims. The most common restriction is on weather events though.
As Geico is a direct company, I'd call up their service number and ask them hypothetically if you were to file a claim from an animal strike if it would impact your rates unfavorably at your next renewal. Depending on who they are, if you say you hit an animal they will set up the claim even if you don't want them to, as it helps them protect their interests.
